I know what you’re saying, but for those new to the concept, careful not to give the wrong impression here. The horse needs to step under the body shadow with the inside hind, but it doesn’t need to step OVER the midline.
Many vaquero-inspired trainers emphasize the value of inside rein to inside hind, or using the inside rein to begin to define where the hind limb moves. Far moreso than inside leg to outside rein (which though incredibly valuable is a finished horse’s aid), this teaches the rider to guide the hindquarters to turn rather than to contain the front end. After all, you can’t really block much of anything with the inside rein in the same way you can with an outside rein.
Not likely to be a trainwreck, per say, but it is the only thing that a dressage rider has to give up in order to make a bridle horse.
Many dressage riders are trained to drop the outside leg back when turning. There’s nothing wrong with that, per say, but as Buck says you’ll never make a bridle horse (or single handed horse) that way. To guide the horse with just the seat and legs, the outside leg has to either stay at the girth or even come forward a bit when turning.
Far too much riding is CONTAINED riding. Though the horse needs to have proper bend in order to turn in correct form, you don’t actually need to use the outside leg back to block the hindquarter from going straight and use the outside rein to block the shoulder from coming straight. Frankly, that’s how a lot of people ride and are taught to ride.
If the horse is following the seat, you don’t even need to scissor your legs to get him to turn, you just offer the feel of turning one way or the other. Buck provides that example as a way of teaching that feel, and Bryan Neubert’s piece here is a masterful glimpse for those wanting to develop seat and leg connection: http://www.eclectic-horseman.com/mercantile/product_info.php?cPath=50&products_id=650
Using the outside rein to CONTAIN anything is completely incorrect as well. Josh Nichol describes that feeling as being in a rowboat full of holes, and needing every finger and toe just to stay afloat. Sort of hard to enjoy a day fishing on the lake that way. If the horse is plowing out the outside shoulder it generally has nothing to do with the outside rein or shoulder, it’s because the thrust from the hindquarter is directed over the outside shoulder. Redirecting that hindquarter by asking the inside hind to make an engaging step under the body shadow rather than a lateral driving step is then much more appropriate than blocking the horse with the outside rein. In the former you’re redirecting the energy, in the latter you’re blocking or bottling up the energy.